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Stochastic Lotka-Volterra system with unbounded distributed delay

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  • In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects these systems significantly, this paper perturbs the Lotka--Volterra system

    $\dot{x}(t)=\mbox{diag}(x_1(t), \cdots, x_n(t))(r+Ax(t)+B\int_{-\infty}^0x(t+\theta)d\mu(\theta))$

    into the corresponding stochastic system

    $dx(t)=\mbox{diag}(x_1(t), \cdots, x_n(t))[(r+Ax(t)+B\int_{-\infty}^0x(t+\theta)d\mu(\theta))dt+\beta dw(t)].$

    This paper obtains one condition under which the above stochastic system has a global almost surely positive solution and gives the asymptotic pathwise estimation of this solution. This paper also shows that when the noise is sufficiently large, the solution of this stochastic system will converge to zero with probability one. This reveals that the sufficiently large noise may make the population extinct.

    Mathematics Subject Classification: Primary: 34K50; 60H10; Secondary: 92D25.


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